Friction Factor Calculator

Background.

Push water through a pipe and part of the pump's effort is lost to friction at the wall, showing up as a pressure drop along the run. The Darcy–Weisbach equation puts a number on it, Δp = f × (L/D) × ρV²/2, and almost all of the physics hides in one dimensionless coefficient, the Darcy friction factor f. Get f right and pipe sizing, pump selection and flow estimates follow; get it 10% wrong and every pressure drop is 10% wrong too.

In turbulent flow f depends on two things. One is the Reynolds number, Re = VD/ν, a measure of how strongly inertia dominates viscosity. The other is relative roughness ε/D, the typical height of the wall's bumps divided by the bore. At moderate Reynolds numbers a thin viscous layer next to the wall smooths over small bumps, and f falls as Re rises. At very high Reynolds numbers that layer thins until the bumps poke through, and f levels off at a value set by roughness alone.

The standard description of that behaviour is the Colebrook equation, the basis of the Moody chart, but it is implicit: f appears on both sides and has to be found by iteration. In 1983 S. E. Haaland published an explicit alternative needing a single logarithm, and textbooks put it within about 2% of Colebrook across the turbulent range — inside the uncertainty of any real pipe's roughness.

The result is the Darcy factor, four times the Fanning factor used in some chemical-engineering texts. Only Reynolds numbers of 4,000 and above are accepted. Below about 2,300 pipe flow is laminar and f = 64/Re exactly; in between, flow flickers between regimes and no formula is dependable. Fittings, non-circular ducts, compressible gas flow and non-Newtonian fluids are also outside its scope.

What is friction factor calculator?

The Darcy friction factor f is the dimensionless coefficient in the Darcy–Weisbach equation that links the frictional pressure drop in a full, circular pipe to its length, its diameter and the kinetic energy per unit volume of the flow, ρV²/2. For fully developed turbulent flow it depends only on the Reynolds number and the relative roughness of the wall, and it is the same number whatever unit system you work in.

Physically it measures how hard the wall drags on the moving fluid. The average shear stress at the wall is τw = f × ρV²/8 — equivalently the Fanning factor, f/4, multiplied by ρV²/2 — so a force balance on a length of pipe turns that wall stress directly into the Darcy–Weisbach pressure drop. Across ordinary turbulent flow the number stays small: the same formula gives about 0.0060 for a smooth pipe at a Reynolds number of 100 million, 0.0404 for a smooth pipe at 4,000, and 0.0719 for a very rough wall with ε/D = 0.05 at 100,000.

How to use this calculator.

1. 01Work out the Reynolds number first: Re = V × D ÷ ν, with V the mean velocity, D the inside diameter and ν the kinematic viscosity. Water near 20 °C has ν close to 1.0 × 10⁻⁶ m²/s; air at the same temperature is about 1.5 × 10⁻⁵ m²/s, so air needs fifteen times the velocity to reach the same Re. The Reynolds number must be between 4,000 and 100 million.
2. 02Look up the wall roughness ε for your pipe material — commercial steel is usually taken as about 0.045 mm — and divide it by the inside diameter in the same unit. Use the actual bore, not the nominal size: a nominal 2-inch Schedule 40 steel pipe is about 52.5 mm inside. Enter 0 to treat the pipe as perfectly smooth, a fair approximation for drawn tubing and plastic.
3. 03Read the Darcy friction factor. If the pressure-drop formula you are using is written with the Fanning factor, divide by 4.
4. 04Put f into Δp = f × (L/D) × ρV²/2, or h = f × (L/D) × V²/(2g) for head loss, to get the friction loss along your straight pipe run.

The formula.

1/√f = −1.8 log₁₀[(ε/D ÷ 3.7)^1.11 + 6.9/Re]. The two terms inside the bracket stand for the two ways a wall resists turbulent flow. The roughness term, (ε/D ÷ 3.7)^1.11, dominates when the bumps on the wall protrude through the viscous sublayer; the 6.9/Re term dominates in smooth pipes, where viscosity governs the layer next to the wall. Their sum is far below 1 in any real pipe, so its logarithm is negative and the right-hand side comes out positive. The constants 1.8, 3.7, 1.11 and 6.9 are Haaland's fitted values, chosen so the explicit form reproduces the implicit Colebrook equation. Squaring the right-hand side and taking the reciprocal gives f. For a fixed roughness, raising Re lowers f until the 6.9/Re term becomes negligible; at ε/D = 0.001 f settles towards 0.0197 in that fully rough limit.

Colebrook's 1939 equation, which Haaland's form imitates, joined two older laws: the smooth-pipe law of Prandtl and von Kármán, in which f depends on Re alone, and the fully rough law from Nikuradse's experiments on pipes lined with uniform sand grains, in which it depends on ε/D alone. That history explains what ε really is. It is not a measured height of the bumps on a real pipe but an equivalent sand-grain roughness — the sand size that would produce the same friction — and the values tabulated for commercial materials were inferred from friction tests.

Against an iterated Colebrook solution, Haaland's version runs about 1.3% high for a smooth pipe at Re = 4,000, about 0.9% low for a smooth pipe at Re = 100,000, and within 0.2% for rough walls with ε/D between 0.01 and 0.05. Outside the range it was fitted to, the equation stops meaning anything. The Moody chart ends at ε/D = 0.05; far beyond that a 'roughness' comparable to the bore no longer describes a pipe, and as (ε/D ÷ 3.7)^1.11 approaches 1 the logarithm heads to zero and the formula returns nonsense, so the calculator refuses a relative roughness above 0.05 and a Reynolds number above 100 million, the edges of the chart.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Haaland, S. E., Simple and Explicit Formulas for the Friction Factor in Turbulent Pipe Flow, Journal of Fluids Engineering 105(1), 1983.
2. [2]Çengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, 4th ed. (PRINT).
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.
4. [4]NIST Special Publication 811, Guide for the Use of the International System of Units, 2008 edition.
5. [5]NIST Chemistry WebBook, fluid-property data context.